Let $Z_n$ be the additive group of modulo $n$. Determine all the value of $n$ such that there exists bijective function $f: Z_n \mapsto Z_n$ and $g: Z_n \mapsto Z_n$, so $f+g: Z_n \mapsto Z_n$ is also bijective function.
I have discussed this with my classmates, but gives uncertainty answer. We found that if $n$ is odd number, then we have that it satisfies the condition. So, my teacher said that we should look for contradiction by letting $n$ is even number. But, how to run this step?